Details of RMFT calculations are explained as follows [1]}. We first introduce
three Gutzwiller renormalization factor to approximate the operator that
projects out the doubly occupied states: \(g_{t}=n_{h}/(1+n_{h})\)  for the
hopping \(t\)  and \(t^{\prime }\) , \(g_{J}=4/(1+n_{h})^{2}\)  for the superexchange
\(J\) , and \(g_{K}=2/(1+n_{h})\)  for the Kondo coupling \(K\) . We then define four
mean-field order parameters to decouple the Heisenberg superexchange and
Kondo terms:
\(\chi _{ij} &=&\langle d_{i\uparrow }^{\dagger }d_{j_{\uparrow }}+d_{i\downarrow }^{\dagger }d_{j\downarrow }\rangle ,\quad B=\frac{1}{\sqrt{2}}\langle d_{j\uparrow }^{\dagger }c_{j\downarrow }^{\dagger }-d_{j\downarrow }^{\dagger }c_{j\uparrow }^{\dagger }\rangle , \\\Delta _{ij} &=&\langle d_{i\uparrow }^{\dagger }d_{j_{\downarrow }}^{\dagger }-d_{i\downarrow }^{\dagger }d_{j\uparrow }^{\dagger }\rangle ,\quad D=\frac{1}{\sqrt{2}}\langle c_{j\uparrow }^{\dagger }d_{j\uparrow }+c_{j\downarrow }^{\dagger }d_{j\downarrow }\rangle .\) 
